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©Curriculum Associates, LLC Copying is not permitted. 209Lesson 20 Area of Composed Figures

Name: Area of Composed Figures

Prerequisite: Separating a Figure to Find the Area

Study the example showing how to decompose a shape to find its area. Then solve problems 1–7.

1 In the example, why is the area of one triangle multiplied by 2?

2 Find the dimensions of one of the shaded triangles from Bev’s pattern. What can you say about the areas of all the shaded triangles in Bev’s pattern? Explain.

3 What is the total area of the 6 shaded triangles?

Show your work.

Solution:

Example

Bev creates the pattern shown. Bev’s pattern is made up of 6 triangles, 2 whole hexagons, and 2 half hexagons. She wants to find the area of one of the hexagons.

To find the area of one hexagon, Bev divides a hexagon into 1 rectangle and 2 congruent triangles. Then she finds the area of each shape.

Area of triangle 5 1 ·· 2 bh 5 1 ·· 2 (6)(2) 5 6 square units

Area of rectangle 5 bh 5 2(6) 5 12 square units

Area ofhexagon

5 2 1

Area oftriangle 2 1

Area ofrectangle

622

2

2

Lesson 20

5 2(6) 1 12 5 24 square units

209209

BThere are two triangles in the hexagon, and their areas are equal.

B

The base is 4 units and the height is 3 units. Because the triangles all have the same

dimensions, the areas will be equal.

M

Area of 1 triangle 5 1 ·· 2 bh 5 1 ·· 2 (4)(3) 5 6 square units

Area of 6 triangles 5 6 • 6 5 36 square units

36 square units

©Curriculum Associates, LLC Copying is not permitted.210 Lesson 20 Area of Composed Figures

Solve.

4 A tile for a mosaic is shaped like the parallelogram shown. Find the area that is covered by 1 tile. Explain.

5 Refer to the trapezoid at the right. Write an equation for the area of the trapezoid, A, in terms of the areas of the triangles and rectangle, T and R.

6 Use your equation to fi nd the area of the trapezoid in problem 5.

Show your work.

Solution:

7 Vasco needs to know the area of the four-pointed star shown for an art project. He knows that all of the triangular “points” are identical. What is the area of the star?

Show your work.

Solution:

6.5 cm

3.8 cm

10.6 cm

T T

R

123 in.

125 in.

1.5 cm

2.4 cm

210

B

19 1 ·· 4 in.2 Possible explanation: A 5 bh 5 3 1 ·· 2 3 5 1 ·· 2 5

7 ·· 2 3 11 ··· 2 5 77 ··· 4 5 19 1 ·· 4 square inches

M

A 5 2T 1 R

M

Possible work:

A 5 2T 1 R

A 5 2[( 1 ·· 2 )(3.4)(6.5)] 1 (3.8)(6.5)

A 5 2(11.05) 1 24.7

A 5 22.1 1 24.7

A 5 46.846.8 square centimeters

C

Area of star 5 4(Area of 1 triangle) 1 Area of square

Area of 1 triangle 5 1 ·· 2 bh 5 1 ·· 2 (1.5)(2.4) 5 1.8

Area of square 5 s • s 5 (1.5)(1.5) 5 2.25

Area of star 5 4(1.8) 1 2.25 5 9.45

9.45 square centimeters

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©Curriculum Associates, LLC Copying is not permitted. 211Lesson 20 Area of Composed Figures

Name: Lesson 20

Dividing a Figure to Find Its Area

Study the example showing how to find the area of a composite figure. Then solve problems 1–7.

1 In the example, why were Area B and Area C multiplied by 4 but Area A was not?

2 How do you know the dimensions of rectangle B?

3 How were the dimensions of right triangle C found?

Example

A STOP sign is a polygon with 8 equal sides. Find the area of the STOP sign.

First, draw lines to divide the figure into polygons whose areas you know how to find. Look for polygons that are identical and label them with the same letter. Sketch one of each of the shapes and label its dimensions. Then find their areas.

Area of square A 5 bh 5 (12.4)(12.4) 5 153.76

Area of rectangle B 5 bh 5 (12.4)(8.8) 5 109.12

Area of right triangle C 5 1 ·· 2 bh 5 1 ·· 2 (8.8)(8.8) 5 38.72

Total area 5 153.76 1 4(109.12) 1 4(38.72) 5 745.12

The area is 745.12 square inches.

STOP

30 in.

C

B B

C

C C

B

A

B

30 in. 12.4 in.

12.4 in.

8.8 in.

8.8 in.

12.4 in. 12.4 in.

12.4 in.

A

8.8 in.

B 8.8 in.

8.8 in.

C

211

There are 4 Area Bs and 4 Area Cs, but there is only 1 Area A.

Possible answer: The length, 12.4 in., is marked on the figure. The width can be found by

subtracting 12.4 from 30 and dividing by 2.

Possible answer: The height, 8.8 in., is marked on the figure. The base can be found by

subtracting 12.4 from 30 and dividing by 2.

B

B

B

©Curriculum Associates, LLC Copying is not permitted.212 Lesson 20 Area of Composed Figures

Solve.

4 Show how you can divide the fi gure at the right into polygons whose areas you know how to fi nd.

5 Find the total area of the fi gure in problem 4.

6 Pedro says that the solution to problem 5 is

78 3 ·· 4 square inches. How did he get that answer?

7 Refer to the fi gure at the right.

a. Show how you can divide the figure at the right into polygons whose areas you know how to find. Then write an equation that represents the total area of the figure, T.

b. Sketch each kind of figure and label its dimensions.

c. Find the total area of the figure.

Show your work.

Solution:

1210 in.

1210 in.

127 in.

127 in.

7 cm

2 cm

7 cm 2 cm

2 cm

3 cm

2 cm

3 cm

212

Possible answer is shown on the figure.

20 square centimeters

Possible work: Area of parallelogram 5 bh 5 1 10 1 ·· 2 2 1 7 1 ·· 2 2 5 78 3 ·· 4 square inches

Area of figure 5 2 1 78 3 ·· 4 2 5 157 1 ·· 2 square inches

Area of square A 5 2(2) 5 4

Area of rectangle B 5 2(3) 5 6

Area of right triangle C 5 1 ·· 2 (2)(2) 5 2

T 5 A 1 2B 1 2C

T 5 4 1 2(6) 1 2(2) 5 20

Possible answer: Pedro forgot to multiply by 2 to account for the two parallelograms.

Possible answer is shown on the figure.

Possible equation: T 5 A 1 2B 1 2C

A

B

C

B C

2 cm

2 cmA3 cm

2 cm

B2 cm

2 cm

C

M

M

M

C

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Practice Lesson 20 Area of Com

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©Curriculum Associates, LLC Copying is not permitted.214 Lesson 20 Area of Composed Figures

Solve.

5 A side of a barn has the shape shown. Use subtraction to fi nd the area of the side of the barn. Draw any lines on the diagram that will help explain your work.

Show your work.

Solution:

6 Jasmine solved problem 5 using the diagram at the

right. She says that an expression for the total area is

(100)(75) 1 (50)(25) 1 1 ·· 2 (25)(25). Do you agree? If so,

show that her expression is equivalent to the one you

found in problem 5. If you disagree, explain her error.

Show your work.

Solution:

7 Find the area of the STOP sign using subtraction. Draw any lines needed on the diagram and explain your thinking.

Show your work.

Solution:

50 ft

75 ft

100 ft

100 ft

STOP

30 in.

30 in. 12.4 in.

12.4 in.

8.8 in.

8.8 in.

50 ft

75 ft

100 ft

100 ft

B

A

CC

214

Let A = area of side of barn, S = area of square, and T = area of triangle.

A 5 S 2 2T

5 100(100) 2 2(0.5)(25)(25)

5 10,000 2 625 = 9,375

Find the area of the 30-in. by 30-in. square.

A 5 (30)(30) 5 900

Find the area of one of the right triangles.

A 5 1 ·· 2 bh 5 1 ·· 2 (8.8)(8.8) 5 38.72

Area of STOP sign 5 900 2 4(38.72) 5 745.12

Jasmine added Area A, Area B, and Area C. But

there are two identical triangles with Area C. She

should have multiplied 1 ·· 2 (25)(25) by 2.

The area of the side of the barn is 9,375 square feet.

Disagree; Jasmine should have multiplied the area of triangle C by 2.

745.12 square inches

M

M

C

©Curriculum Associates, LLC Copying is not permitted. 213Lesson 20 Area of Composed Figures

Name: Lesson 20

Solving Problems Related to Area

Study the example showing how to use what you have learned to solve problems related to area. Then solve problems 1–7.

1 What is the area of the large rectangle? Show your work.

2 Find the dimensions of rectangle A. Then fi nd the area of rectangle A. Show your work.

3 Explain how you can fi nd the dimensions of triangle B. Then fi nd the area of triangle B.

4 Find the area of the parking lot. Explain your calculation.

Example

Ben is paving a parking lot with the shape shown. How can he find the area of the parking lot?

One way is to complete a rectangle around the parking lot and then subtract those areas that are not part of the parking lot. This is what Ben decides to do.

Area ofparking lot

5 Area of large

rectangle 2 2(Area A) 2 Area B

100 yd

150 yd

40.5 yd

AB

A

25 yd75 yd

213

11,250 square yards; A 5 ℓw 5 (150)(75) 5 11,250

The length is 40.5 yards and the width is 1 ·· 2 (75 2 25) 5 25, or 25 yards. A 5 ℓw 5 (40.5)(25) 5

1,012.5; 1,012.5 square yards

The base is 150 2 100 2 40.5 5 9.5, or 9.5 yards; the height is 75 2 2(25) 5 25 yards.

A 5 1 ·· 2 bh 5 1 ·· 2 (9.5)(25) 5 118.75; 118.75 square yards

Area of parking lot 5 11,250 2 2(Area A) 2 (Area B)

5 11,250 2 2(1,012.5) 2 118.75 5 9,106.25 square yards

I subtracted 2 times Area A because there are 2 identical rectangles labeled A, as well as

the area of triangle B from the area of the large rectangle.

B

B

B

B

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©Curriculum Associates, LLC Copying is not permitted.216 Lesson 20 Area of Composed Figures

6 What percent of the total area of the fl ag is white?

Show your work.

Solution:

Solve.

4 Write the area of each region of the design in square units.

Area A

Area B

Area C

Total Area D

Total Area

5 Abe plans to lay tiles on the areas shown. A case of tiles costs $40 and covers 15 square feet. How much does Abe need to budget for the tiles?

A $360 C $440

B $400 D $480

Claire chose B for the answer. What did she do wrong?

Solution:

Can Abe buy part of a case?

Did you answer the question asked?

How can you find the length of a line segment on a grid?

8 ft

6 ft

10 ft

12.75 ft

4 ft

8 ft8 ft

12.75 ft

D D

A

B

C

8 in.

4 in. 6 in.

6 in.

2 in.

3 in.

216

Abe needs about 10.8 cases of tiles. Claire multiplied $40 by 10

instead of 11.

50% of the flag is white.

Total area 5

ℓw 5 (8)(6) 5 48

Area of 2 black triangles 5 2 1 1 ·· 2 bh 2 5 2 1 1 ·· 2 2 (4)(3) 5 12

Area of 2 blue triangles 5 2 1 1 ·· 2 bh 2 5 2 1 1 ·· 2 2 (6)(2) 5 12

White area 5 48 2 12 2 12 5 24; Percent white 5 24 ··· 48 5 50%

3

18

2

2

25

M

M

C

©Curriculum Associates, LLC Copying is not permitted. 215Lesson 20 Area of Composed Figures

Name:

1 Complete the equations by writing A, B, C, or D in each box to represent the area of each region. The large rectangle is D.

Area of trapezoid 5 1 2 3

Area of trapezoid 5 2 2 3

Area of Composed Figures

Solve the problems.

3 Perry drew this design on graph paper. He says that because the design is made of 2 identical squares he can fi nd the area of his design by multiplying the area of one square by 2. Do you agree? If so, explain why. If not, explain Perry’s error. Draw lines on the design to show your thinking.

2 The dimensions of a small fl ag are shown. Which of the following expressions does NOT represent the area of the region shown in white?

A (2)(6) 2 1 ·· 2 (2)(3)

B (4)(6) 2 1 ·· 2 (4)(3) 2 2(3) 2 1 ·· 2 (3)(2)

C (3)(2) 1 1 ·· 2 (3)(2)

D (6)(2) 1 1 ·· 2 (3)(4)

What are two different ways that you could find the area of the trapezoid?

How many ways can you think of to find the area of the white region?

How would you find the area of the design?

Lesson 20

AB

CC

B

4 in.

6 in.

2 in.

3 in.3 in.

215

Disagree; Possible answer: Because the two squares overlap, Perry needs to subtract

the area of the part that is underneath the top square from twice the area of 1 square.

A

D

B

C

B

M

M